The signless Laplacian spectral Tur\'an problems for hypergraphs

TL;DR

This paper extends spectral Turán results to hypergraphs using the signless Laplacian spectral radius, providing a general theorem and applying it to determine the extremal hypergraph avoiding Fano planes.

Contribution

It introduces a general theorem linking degree-stability and spectral Turán problems for hypergraphs, extending prior results to the signless Laplacian spectral setting.

Findings

Established a general theorem for spectral Turán problems in hypergraphs.

Determined the extremal hypergraph avoiding Fano planes as the balanced complete bipartite hypergraph.

Extended spectral Turán results to the setting of the signless Laplacian spectral radius.

Abstract

Let $\mathcal{H}=(V, E)$ be an $r$-uniform hypergraph on $n$ vertices. The signless Laplacian spectral radius of $\mathcal{H}$ is defined as the maximum modulus of the eigenvalues of the tensor $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree diagonal tensor and the adjacency tensor of $\mathcal{H}$, respectively. In this paper, we establish a general theorem that extends the spectral Tur\'an result of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)] to the setting of signless Laplacian spectral Tur\'an problems. We prove that if a family $\mathcal{F}$ of $r$-uniform hypergraphs is degree-stable with respect to a family $\mathcal{H}_n$ of $r$-uniform hypergraphs and its extremal constructions satisfy certain natural assumptions, then the signless Laplacian spectral Tur\'an problem for $\mathcal{F}$ can be effectively reduced to the corresponding problem restricted to the family $\mathcal{H}_n$. As a concrete application, we completely determine the extremal hypergraph that maximizes the signless Laplacian spectral radius among all Fano plane-free $3$-uniform hypergraphs, showing that the unique extremal hypergraph is the balanced complete bipartite $3$-uniform hypergraph.